mean_and_variance_of_binomial_distribution
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| mean_and_variance_of_binomial_distribution [2023/10/18 16:05] – hkimscil | mean_and_variance_of_binomial_distribution [2024/10/14 17:06] (current) – [For variance] hkimscil | ||
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| ====== For Mean ====== | ====== For Mean ====== | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | E(X) & = & \sum_{x}x p(x) \\ | + | E(X) & = & \sum_{x}x p(x) \;\;\; \because \; p(x) = {{n}\choose{x}} p^x (1-p)^{n-x} |
| - | & = & \sum_{x=0}^{n} x {{n}\choose {x}} p^x(1-p)^{n-x} | + | & = & \sum_{x=0}^{n} x {{n} \choose {x}} p^x(1-p)^{n-x} |
| & = & \sum_{x=0}^{n} x \frac{n!}{x!(n-x)!} p^x(1-p)^{n-x} | & = & \sum_{x=0}^{n} x \frac{n!}{x!(n-x)!} p^x(1-p)^{n-x} | ||
| - | \text{note that } x! = x(x-1)! \\ | + | \text{note that } x! = x(x-1)! |
| & = & \sum_{x=1}^{n} \frac{n!}{(x-1)!(n-x)!} p^x(1-p)^{n-x} | & = & \sum_{x=1}^{n} \frac{n!}{(x-1)!(n-x)!} p^x(1-p)^{n-x} | ||
| \text{cause we know that E(x) = np,} \\ | \text{cause we know that E(x) = np,} \\ | ||
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| \text{and | \text{and | ||
| & = & \sum_{x=1}^{n} \frac{\underline{n}*(n-1)!}{(x-1)!(n-x)!} (\underline{p}*p^{x-1})(1-p)^{n-x} | & = & \sum_{x=1}^{n} \frac{\underline{n}*(n-1)!}{(x-1)!(n-x)!} (\underline{p}*p^{x-1})(1-p)^{n-x} | ||
| - | \text{we take out the underlined part} \\ | + | \text{we take out the underlined part} \\ |
| \text{(that is, np) out of the sigma part } \\ | \text{(that is, np) out of the sigma part } \\ | ||
| & = & np \sum_{x=1}^{n} \frac{(n-1)!}{(x-1)!(n-x)!} p^{x-1}(1-p)^{n-x} | & = & np \sum_{x=1}^{n} \frac{(n-1)!}{(x-1)!(n-x)!} p^{x-1}(1-p)^{n-x} | ||
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| & = & np \sum_{y=0}^{m} \frac{(m)!}{(y)!(m-y))!} p^{y}(1-p)^{(m-y)} | & = & np \sum_{y=0}^{m} \frac{(m)!}{(y)!(m-y))!} p^{y}(1-p)^{(m-y)} | ||
| & = & np \sum_{y=0}^{m} {{m}\choose {y}} p^{y}(1-p)^{(m-y)} | & = & np \sum_{y=0}^{m} {{m}\choose {y}} p^{y}(1-p)^{(m-y)} | ||
| - | \text{Recall that} \\ | + | \text{Recall that} \\ |
| \sum^{m}_{y=0}{{m}\choose{y}} a^{y} b^{m-y} = (a + b)^{m} \\ | \sum^{m}_{y=0}{{m}\choose{y}} a^{y} b^{m-y} = (a + b)^{m} \\ | ||
| - | & = & np (p + (1-p))^m | + | & = & np\;(p + (1-p))^m |
| - | & = & np (1)^m \\ | + | & = & np\;(1)^m \\ |
| & = & np \\ | & = & np \\ | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
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| \begin{align*} | \begin{align*} | ||
| - | E(X^2) & = \sum_{x=0}^{n} x^2 {{n}\choose{x}} p^x (1-p)^{n-x} \\ | + | E(X^2) & = \sum_{x=0}^{n} x^2 p(x) \\ |
| + | & = \sum_{x=0}^{n} x^2 {{n}\choose{x}} p^x (1-p)^{n-x} \;\;\; \because \; p(x) = {{n}\choose{x}} p^x (1-p)^{n-x} \\ | ||
| & = \sum_{x=0}^{n} x^2 \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x} \\ | & = \sum_{x=0}^{n} x^2 \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x} \\ | ||
| \end{align*} | \end{align*} | ||
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| \begin{align*} | \begin{align*} | ||
| E[X(X-1)] & = \sum_{x=0}^{n} x(x-1) \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x} \\ | E[X(X-1)] & = \sum_{x=0}^{n} x(x-1) \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x} \\ | ||
| - | & = \sum_{x=2}^{n} \frac{n!}{(x-2)!(n-x)!} p^x (1-p)^{n-x} \\ | + | & = \sum_{x=2}^{n} \frac{n!}{(x-2)!(n-x)!} p^x (1-p)^{n-x} |
| & = n(n-1)p^2 \sum_{x=2}^{n} \frac{(n-2)!}{(x-2)!(n-x)!} p^{x-2} (1-p)^{n-x} \\ | & = n(n-1)p^2 \sum_{x=2}^{n} \frac{(n-2)!}{(x-2)!(n-x)!} p^{x-2} (1-p)^{n-x} \\ | ||
| \text{cause} \\ | \text{cause} \\ | ||
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| \text {we know that the underline part is} \\ | \text {we know that the underline part is} \\ | ||
| + | (p+(1-p))^m \\ | ||
| + | \text {and, we also know that it is 1} \\ | ||
| (p+(1-p))^m = 1^m \\ | (p+(1-p))^m = 1^m \\ | ||
| & = n(n-1)p^2 (p + (1-p))^m \\ | & = n(n-1)p^2 (p + (1-p))^m \\ | ||
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| E[X(X - 1)] & = n(n-1)p^2 \\ | E[X(X - 1)] & = n(n-1)p^2 \\ | ||
| E[X^2 - X] & = n(n-1)p^2 \\ | E[X^2 - X] & = n(n-1)p^2 \\ | ||
| - | E[X^2]- E[X] & = n(n-1)p^2 \\ | + | E[X^2]- E[X] & = n(n-1)p^2 |
| - | E[X^2]- np & = n(n-1)p^2 \\ | + | E[X^2]- np & = n(n-1)p^2 |
| E[X^2]& = n(n-1)p^2 + np \\ | E[X^2]& = n(n-1)p^2 + np \\ | ||
| \\ | \\ | ||
mean_and_variance_of_binomial_distribution.1697612701.txt.gz · Last modified: 2023/10/18 16:05 by hkimscil